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Frequency and amplitude dependence of the axial and radial stiffness of carbon-black filled rubber bushings
ARTICLE IN PRESS
POLYMER TESTING Polymer Testing 26 (2007) 629–638 www.elsevier.com/locate/polytest
Data Prediction
Frequency and amplitude dependence of the axial and radial stiffness of carbon-black filled rubber bushings M.J. Garcı´ a Ta´rragoa,b,, L. Karia, J. Vinolasb, N. Gil-Negreteb a
The Marcus Wallenberg Laboratory for Sound and Vibration Research, Department of Aeronautical and Vehicle Engineering, KTH, 100 44 Stockholm, Sweden b Department of Applied Mechanics, Ceit and Tecnun (University of Navarra), Manuel de Lardiza´bal 15, 20018 San Sebastia´n, Spain Received 6 February 2007; accepted 26 March 2007
Abstract The frequency and amplitude dependent dynamic behavior of carbon-black filled rubber bushings is experimentally investigated for a commercially available bushing in the axial and radial directions. Based on measurement observations, models for the axial and radial dynamic stiffness of rubber bushings are developed. The amplitude dependence—referred to as the Fletcher–Gent effect and mainly caused by the presence of carbon-black fillers in the rubber—is included in the analytical models by means of equivalent shear moduli, which result from applying a separable elastic, viscoelastic and friction material model to equivalent strains of the non-homogeneous strain states inside the bushing when subjected to axial or radial deflections. Good correlations between measurements and the axial and radial models at amplitudes of 0.1, 0.2 and 0.5 mm from 5 to 155 Hz—when the material parameters are achieved from axial measurements at 0.1 mm—prove the accuracy of both stiffness models. r 2007 Elsevier Ltd. All rights reserved. Keywords: Rubber bushing; Carbon black; Fletcher–Gent effect; Equivalent shear modulus; Dynamic stiffness
1. Introduction Rubber is a versatile polymer employed in many vehicle applications, for instance its elasticity and inherent damping make rubber bushings interesting for use in automotive suspensions. Furthermore, for engineering applications the rubber usually contains fillers, generally one of the many kinds of carbon black. This results in amplitude dependence of the dynamic behavior of the rubber—referred to as the Fletcher–Gent effect [1]. An intact filler structure Corresponding author. Tel.: +34 943 212800; fax: +34 943 213076. E-mail address: [email protected] (M.J. Garcı´ a Ta´rrago).
0142-9418/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2007.03.011
displays a large shear modulus magnitude for small amplitudes due to the interactions within the filler and between the filler and the rubber matrix, but as the amplitude increases the structure links break resulting in modulus decrease. Furthermore, the breaking of filler structure, described as frictional behavior, initially increases the loss factor, but as the amplitude increases further the frictional behavior is released, resulting in a low loss factor. This nonlinear behavior, extensively investigated by Medalia [2,3], Payne [4], Dean et al. [5], Jurado et al. [6], Sommer and Meyer [7] and Wang [8], has been characterized by several authors. Some of them developed micromechanically motivated material models, such as Kraus [9] who explains the
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amplitude dependence as due to the continuous breaking and reforming of van der Waals bonds between carbon-black aggregates. Modifications and experimental application were carried out by Ulmer [10] and Lion [11] presenting a time domain formulation of the Kraus model. Other authors use frictional models, such as Gregory [12], Coveney et al. [13] and Kaliske and Rothert [14] proposing the Prandtl element, a Coulumb damper in series with an elastic spring; a model expanded by Bruni and Collina [15], Olsson and Austrell [16] and Brackbill et al. [17]. Dynamic measurements performed on a commercially available rubber bushing reveal that the component presents frequency and amplitude dependence similar to that of the shear modulus of carbon-black filled rubber [3]. However, the previously described material models cannot be directly applied when modelling the dynamic stiffness of rubber bushings because the strain state inside the deformed components is not homogeneous. Some authors employ material models with finite element simulations and calculate the dynamic stiffness after long-time overlay procedures, such as Austrell et al. [18–20] who represent frequency and amplitude dependence by adding integer derivatives to stickslip friction components, similar to Gil-Negrete [21] except for the use of fractional derivatives. On the other hand, other authors work directly at the component level, like Berg [22] who presents a fiveparameter model that gives good resemblance to the rubber frictional behavior, also used by Sjo¨berg and Kari [23] together with a rate-dependent part using fractional derivatives. Misaji et al. [24] took into account amplitude dependence with the parameters of an ordinary Kelvin–Voigt model updated continuously for every oscillation cycle. However, the latter methods neglect the complicated amplitude dependence inside the rubber bushing due to nonhomogeneous strain states. This paper verifies the accuracy of novel engineering models representing the dynamic behavior of carbon-black filled rubber bushings in axial and radial directions while including frequency and amplitude dependence. A novel model for the radial stiffness is developed here while the formula for the axial stiffness—also applied in this paper—was previously introduced in Garcı´ a Ta´rrago et al. [26]. The analytical models are based on measurement observations and depend on the bushing geometry and an equivalent shear modulus. The latter is obtained by applying a separable elastic,
viscoelastic and friction material model [25,26] to an equivalent strain of the non-homogeneous strain state inside the bushing when subjected to axial or radial deflections. The good agreement between measured and modelled stiffness in axial and radial directions at amplitudes of 0.1, 0.2 and 0.5 mm over a frequency range from 5 to 155 Hz when the model parameters are achieved from axial measurements at 0.1 mm proves the accuracy of the stiffness models. 2. Experimental The frequency and amplitude dependence of carbon-black filled rubber bushings was experimentally investigated for a commercially available bushing. Measurements were carried out in a servo-hydraulic testing machine. The test object in Fig. 1, with mean length L ¼ 34 mm, and inner and outer radii a ¼ 10 mm and b ¼ 18 mm, respectively, was manufactured by Caucho Metal Productos (CMP) under the trade name 15 961 with rubber hardness, Shore A of 61 2. The vulcanized compound contains NR (natural rubber) and 27 phr (parts per hundred rubber) of reinforcing carbon-black filler. 2.1. Setup Experiments were conducted in three stages, see Fig. 2. Firstly, the test object was mounted on a Schenck Series 56 servo-hydraulic test machine whose main components are the frame, an actuator and a hydraulic power pack. The frame consists of the smooth standard platen, columns and the crosshead. The actuator, mounted on the crosshead, is a double-acting force generator, having equal piston areas in the two cylinder chambers. Inside
Fig. 1. Rubber bushing manufactured by Caucho Metal Productos (CMP) under the trade name 15961.
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SCHENCK SERVO-HYDRAULIC TEST MACHINE
LOAD F FEEDBACK SIGNAL PROCESSOR
SENSOR
ADC
DISPL.
d
SENSOR
ADC
RUBBER BUSHING
i
DIALOGUE PROCESSOR
CONTROL PROCESSOR
DAC
PISTON
SERIAL INTERFACE
INTERFACE
F : Force d : Displacement i : Current ADC : Analog-to-digital converter DAC : Digital-to-analogconverter
Fig. 2. Measurement setup.
each piston, one inductive displacement transducer measures the piston position and the piston displacement. The actuator force corresponds to the difference in the oil pressure of the two actuator chambers, measured by pressure transducers based on a strain gauge measuring system. The resulting force in the bushing is measured by a load sensor. Next, measured force and displacement data are sent to a Hydropuls electronic unit Series 56 control panel—which can constantly digitize up to five measuring channels. The signal to the actuator is then automatically adjusted in a control loop with regard to the control displacement signal. Finally,
the data are processed and transmitted to the PC where the frequency signals are displayed. No external analyzer is needed. 2.2. Measurements Dynamic measurements were performed on the test object at displacement amplitudes of 0.1, 0.2 and 0.5 mm in axial and radial directions. The bushing in Fig. 1 was excited by stepped sine displacements starting at 5 Hz and increasing with a constant frequency step of 10 Hz to a maximum frequency of 155 Hz, with amplitude held constant.
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632
Magnitude (N/m)
8
Axial
x 105
x 105
8
6
6
4
4 0.1mm 0.2mm 0.5mm
2
25 Hz 75 Hz 115 Hz
2
0
0 50
100
150
1
2
3
4
5
Loss Factor
x 10-4 0.15
0.15
0.1
0.1
0.05
0.1 mm 0.2 mm 0.5 mm
25 Hz 75 Hz 115 Hz
0.05
0
0 50 100 Frequency (Hz)
150
1
2
3 4 Amplitude (m)
5 x
10-4
Fig. 3. Left: Frequency dependence of measured axial stiffness at three amplitudes. Right: Amplitude dependence of measured axial stiffness at three frequencies.
The resulting complex stiffness in Figs. 3 and 4 is expressed by a magnitude and a loss factor—defined as the tangent of the loss angle. Results reveal that the stiffness presents frequency and amplitude dependence similar to that of the shear modulus of carbon-black filled rubber. The magnitude and loss factor increase with frequency, as shown on the left of Figs. 3 and 4, while the magnitude takes high values for small amplitudes, but as the amplitude increases the magnitude decreases. Furthermore, the loss factor initially increases but as the amplitude increases further it decreases, as displayed on the right of Figs. 3 and 4. The drop of the radial loss factor—measured at 0.5 mm of amplitude—over the frequency range is mainly attributed to the difficulties in accurately measuring amplitudes higher than 0.2 mm. Finally, quasi-static measurements were performed on the servo-hydraulic Schenck testing machine using the same setup. 3. Model Based on measurement observations, analytical models for the axial and radial stiffness were
developed. The former was already presented in Garcı´ a Ta´rrago et al. [26] while a novel radial model is expounded here. They are functions of the bushing dimensions and an equivalent shear modulus. The latter was obtained by applying a newly developed rubber material model to an equivalent strain of the non-homogeneous strain state inside the rubber bushing when subjected to axial or radial deformations. 3.1. Rubber model The rubber material model recently developed in Garcı´ a Ta´rrago et al. [25] is the result of applying the force–displacement relation of the rubber component model established by Sjo¨berg and Kari [23] to the stress–strain level. It consists of three parallel components which represent the elasticity, viscoelasticity and amplitude dependence by using only five parameters. Firstly, in Fig. 5 the elasticity is modelled by a linear relation between the elastic stress selast and strain ; selast ðtÞ ¼ 2mðtÞ,
(1)
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Radial
Magnitude (N/m)
x 106 6
6
4
4
2
x 106
2
0.1mm 0.2mm 0.5mm
0
633
25 Hz 75 Hz 115 Hz
0 50
100
1
150
2
3
4
5
Loss Factor
x 10-4
0.1
0.1
0.1 mm 0.2 mm 0.5 mm
0.05
0
25 Hz 75 Hz 115 Hz
0.05
0 50 100 Frequency (Hz)
150
1
2
3 4 Amplitude (m)
5 x 10-4
Fig. 4. Left: Frequency dependence of measured radial stiffness at three amplitudes. Right: Amplitude dependence of measured radial stiffness at three frequencies.
σtotal (t)
where sfract is the viscoelastic stress and Da denotes the fractional time derivative of order a, defined through an analytical continuation of a fractional Riemann–Liouville integration [28]. Numerically, the viscoelastic component is evaluated as
ε (t)
m, α 2μ
σf max, ε1/2
sfract ðtÞ m
Fig. 5. Rubber material model.
where m is the elastic shear modulus and t is time. The second branch in Fig. 5 indicates the frequency dependence and is modelled by using fractional derivatives which increase the ability to fit to measured frequency dependent characteristics while keeping the number of parameters to only two: a proportionality constant m and the time derivative order a; sfract ðtÞ ¼ mDa ðtÞ;
0oap1,
(2)
n1 Dta X Gðj aÞ nj , GðaÞ j¼0 Gðj þ 1Þ
(3)
where nj ¼ ððn jÞ DtÞ, n is the final strain at time tn ¼ n Dt, Dt is a constant time step applied in the estimation process and G denotes the Gamma function [29] defined as Z 1 1 X ð1Þj , GðbÞ ¼ xb1 ex dx þ j!ðb þ jÞ 1 j¼0 ba0; 1; 2; . . . .
ð4Þ
The third branch in the rubber model represents the amplitude dependence by a smooth friction component [22] which enables a very good fit to measured curves using only two parameters sfmax and 1=2 . The frictional stress sfrict develops gradually
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following the equation: sfrict ðtÞ ¼ sfs þ
½ðtÞ s ½sfmax signð_Þsfs , 1=2 ½1 signð_Þsfs =sfmax þ signð_Þ½ðtÞ s
(5) where the material parameters sfmax and 1=2 are the maximum possible friction stress and the strain needed to develop half of that stress, respectively, with signð_Þ denoting the sign of the strain rate. The values of sfs and s are updated each time the strain changes direction at _ ¼ 0 as sfs sfrict j_¼0 and s j_¼0 . Finally, the rubber behavior is represented by a total stress which is the sum of the three stresses: stotal ðtÞ ¼ selast ðtÞ þ sfract ðtÞ þ sfrict ðtÞ.
(6)
However, a shear modulus cannot be directly achieved from the non-linear relation in Eq. (6) between the total stress and strain, as would be the case with the elastic component, Eq. (1). Moreover, that relation is valid at that specific strain, but in order to represent the behavior of the whole bushing a ‘global’ value for the shear modulus is required. A simplified and effective solution consists in calculating an equivalent shear strain of the nonhomogeneous strain state inside the bushing by using the classical theory of elasticity. Next, the rubber material model is applied to it, thus yielding an equivalent shear modulus which is inserted into an analytical formula for the bushing stiffness.
and d a the amplitude. Similarly, a harmonic radial displacement d r ðtÞ ¼ d r sinðo0 tÞ applied at the outer surface of the bushing results in an equivalent strain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 equiv r ðtÞ ¼ d r ðtÞ . 4 ½b a4 loge ðb=aÞ ½a2 b2 2 (8) 3.3. Stiffness formulas The separable elastic, viscoelastic and friction material model presented in Section 3.1 is applied to the time-domain equivalent strains in Eqs. (7) and (8), thus yielding total stresses stotal a=r ðtÞ of the form displayed in Eq. (6). Next, an equivalent shear modulus is obtained by dividing the temporal Fourier transform (~) of the total stress s~ total a=r ðoÞ by the corresponding equivalent strain ~ equiv a=r ðoÞ Axial 500 400 300 200 Force (N)
634
100 0 -100 -200
3.2. Equivalent strains
-300
The strain states inside the rubber bushing in Fig. 1 when subjected to axial and radial deflections are obtained with the classical linear theory of elasticity while considering that the constitutive equations between the stress and strain contain only elastic and frequency dependent components, and that the wave effects within the bushing are negligible for the frequency range of interest. The amplitude dependence is taken into account later when the rubber model is applied to the equivalent strains. An energy density balance [25] between the rubber bushing deformed axially and a simple shear specimen made of the same material leads to the axial equivalent strain d a ðtÞ equiv a ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 2½b a2 loge ðb=aÞ
(7)
where d a ðtÞ ¼ d a sinðo0 tÞ is the harmonic axial displacement with o0 the angular excitation frequency
Model
-400
Measurement
-500 -1
-0.5
0 Displacement (m)
0.5
1 x
10-3
Fig. 6. Quasi-static measured and modelled hysteresis loop at amplitude of 1 mm and frequency of 0.1 Hz.
Table 1 Optimized material parameters for the bushing in Fig. 1 obtained from axial measurements Material parameters m ¼ 1:09 106 N=m2 m ¼ 2:0 105 Nsa =m2 a ¼ 0:24 sf max ¼ 1:46 104 N=m2 1=2 ¼ 1:3 102
ARTICLE IN PRESS M.J. Garcı´a Ta´rrago et al. / Polymer Testing 26 (2007) 629–638
x 105
evaluated at the excitation frequency o0 ,
Axial
Magnitude (N/m)
8
m^ equiv
6
Mod. 0.1mm Meas. 0.1mm
2 0 40
60
80
100
120
140
Loss Factor
0.15 0.1
k^dyn
0.05
Mod. 0.1mm Meas. 0.1mm 40
60 80 100 Frequency (Hz)
120
a
¼
140
s~ total a=r ðo0 Þ . 2 ~equiv a=r ðo0 Þ
(9)
2pL m^ , loge ðb=aÞ equiv a
(10)
10pL m^ equiv r , 2 2 2 2 2 loge ðb=aÞ ½ðb a =b þ a Þ þ D
k^dyn r ¼ 7
Fig. 7. Measured and modelled axial stiffness versus frequency at amplitude of 0.1 mm.
6
3 2
(11) Axial
x 105
Magnitude (N/m)
¼
while the radial equivalent modulus is inserted into the formula for the radial stiffness of a bushing of finite length developed by Horton et al. [27]:
0 20
a=r ðo0 Þ
This linearization process takes into account the non-linear relation between stress and strain at frequency o0 considering the first order response while omitting the less important overtones (stress response at 3o0 , 5o0 ; . . . ). Finally, the axial equivalent modulus is inserted into the formula for the axial stiffness calculated previously with the classical theory of elasticity
4
20
635
x 105
6
4
4 Mod. 0.2mm Meas. 0.2mm Mod. 0.5mm Meas. 0.5mm
2
Mod. 45 Hz Meas. 45 Hz Mod. 115 Hz Meas. 115 Hz
2
0
0 50
100
150
1
2
3
4
5
Loss Factor
x 10-4
0.15
0.15
0.1
0.1
0.05
Mod. 0.2mm Meas. 0.2mm Mod. 0.5mm Meas. 0.5mm
Mod. 45 Hz Meas. 45 Hz Mod. 115 Hz Meas. 115 Hz
0.05
0
0 50 100 Frequency (Hz)
150
1
2
3 Amplitude (m)
4
5 x 10-4
Fig. 8. Left: Measured and modelled axial stiffness versus frequency at two amplitudes. Right: Measured and modelled axial stiffness versus amplitude at two frequencies.
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section to verify the accuracy of Eqs. (10) and (11) modelling the axial and radial stiffness of carbonblack filled rubber bushings. Firstly, the model parameters corresponding to the analyzed bushing are determined. The dimensions of the bushing are straightforwardly obtained while the material parameters—common to both models—are extracted only from axial tests. The elastic and amplitude dependence parameters m, sfmax and 1=2 are those which optimize the agreement between measured and modelled hysteresis loops for the axial case at amplitude of 1 mm and frequency of 0.1 Hz, as shown in Fig. 6. The frequency dependence parameters m and a are obtained from a minimization of the errors between the axial model and measurements at amplitude of 0.1 mm over a frequency range from 5 to 155 Hz. The optimized material parameters values are in Table 1 with the corresponding stiffness displayed in Fig. 7. Next, the axial model—using the parameters values in Table 1—is carried out at amplitudes of 0.2 and 0.5 mm from 5 to 155 Hz and the results are
with 4ðb2 þ a2 Þ abðb2 þ 3a2 Þ½I 1 ðabÞK 0 ðaaÞ þ I 0 ðaaÞK 1 ðabÞ D¼ a2 baðb2 þ a2 Þ½I 1 ðabÞK 1 ðaaÞ þ I 1 ðaaÞK 1 ðabÞ
aað3b2 þ a2 Þ½I 1 ðaaÞK 0 ðabÞ þ I 0 ðabÞK 1 ðaaÞ , a2 baðb2 þ a2 Þ½I 1 ðabÞK 1 ðaaÞ þ I 1 ðaaÞK 1 ðabÞ
ð12Þ where I n ðarÞ and K n ðarÞ are modified Bessel functions of the first and second kinds, respectively, and of order n [29], r is the radius and a2 ¼ 60=L2 . These engineering formulas represent the dynamic behavior including frequency and amplitude dependence of a carbon-black filled rubber bushing in axial and radial directions. The complex stiffness is expressed by a magnitude and a loss factor, the latter defined as the tangent of the phase between stress and strain at the excitation frequency. 4. Results and discussions Measurements performed on the commercially available bushing in Fig. 1 are employed in this
Radial
Magnitude (N/m)
x 106
x 106
5
5
4
4
3
3 Mod. 0.1mm Meas. 0.1mm Mod. 0.2mm Meas. 0.2mm
2 1
Mod. 25 Hz Meas. 25 Hz Mod. 75 Hz Meas. 75 Hz
2 1 0
0 50
100
150
1
2
3
4
5 x 10-4
Loss Factor
0.15
0.1
0.1
0.05
Mod. 0.1mm Meas. 0.1mm Mod. 0.2mm Meas. 0.2mm
Mod. 25 Hz Meas. 25 Hz Mod. 75 Hz Meas. 75 Hz
0.05
0
0 50 100 Frequency (Hz)
150
1
2
3 4 Amplitude (m)
5 x 10-4
Fig. 9. Left: Measured and modelled radial stiffness versus frequency at two amplitudes. Right: Measured and modelled radial stiffness versus amplitude at two frequencies.
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compared with the dynamic measurements presented in Section 2, showing good agreement, as seen in Fig. 8. Finally, the radial model—making use of identical parameter values as previously employed in the axial model—is performed at amplitudes of 0.1, 0.2 and 0.5 mm and the results are displayed together with the experimental data, showing the good agreement presented in Fig. 9. Consequently, the effectiveness of Eqs. (10) and (11) modelling the global behavior of filled rubber bushings including frequency and amplitude dependence is verified.
637
Unlike previous models, these formulas implemented in MATLABs are simple and fast engineering tools to predict the dynamic stiffness of rubber bushings including frequency and amplitude dependence. The simplicity—characteristic of these models—is important when a rubber bushing is only a small component in more complex systems where the dynamic behavior is to be analyzed. Acknowledgment Caucho Metal Productos (CMP) is gratefully acknowledged for providing the sample bushings.
5. Conclusions This paper experimentally verifies the accuracy of novel engineering models representing the dynamic behavior of carbon-black filled rubber bushings in axial and radial directions, while including frequency and amplitude dependence. The model for the radial stiffness is developed in this paper while the axial stiffness model was previously introduced in Garcı´ a Ta´rrago et al. [26]. Experiments were carried out on a commercially available bushing. Good agreement between measured and modelled axial and radial stiffness at amplitudes of 0.1, 0.2 and 0.5 mm over a frequency range from 5 to 155 Hz—when the material parameters are achieved from axial measurements at 0.1 mm—prove the accuracy of both stiffness models. Therefore, these engineering formulas together with the torsion model developed in Garcı´ a Ta´rrago et al. [25] represent accurately enough the global dynamic behavior of filled rubber bushings including frequency and amplitude dependence. The developed analytical models are based on observations of the measurements performed on the carbon-black filled rubber bushing, showing that the frequency and amplitude dependence of its dynamic stiffness is similar to that dependence of the shear modulus of that type of rubber. Thus, the engineering formulas are function of the bushing dimensions and an equivalent shear modulus. The latter is achieved by applying a separable elastic, viscoelastic and friction material model to an equivalent strain of the non-homogeneous strain state inside the bushing when deformed. The rubber model is the result of applying a force–displacement relation of a well established rubber component model [23]—experimentally validated on commercial bushings—to the stress–strain level.
References [1] W.P. Fletcher, A.N. Gent, Non-linearity in the dynamic properties of vulcanised rubber compounds, I.R.I. Transactions 29 (1953) 266–280. [2] A.I. Medalia, G. Kraus, Reinforcement of elastomers by particulate fillers, in: J.E. Mark, B. Erman, F.R. Eirich (Eds.), Science and Technology of Rubber, Academic Press, San Diego, 1994, pp. 387–418. [3] A.I. Medalia, Effects of carbon black on dynamic properties of rubber, Rubber Chem. Technol. 51 (1978) 437–523. [4] A.R. Payne, R.E. Whittaker, Low strain dynamic properties of filled rubbers, Rubber Chem. Technol. 44 (1971) 440–478. [5] G.D. Dean, J.C. Duncan, A.F. Johnson, Determination of nonlinear dynamic properties of carbon-filled rubbers, Polym. Test. 4 (2–4) (1984) 225–249. [6] F.J. Jurado, A. Mateo, N. Gil-Negrete, J. Vinolas, L. Kari, Testing and FE modelling of the dynamic properties of carbon black filled rubber, in: Proceeding of EAEC 99, Barcelona, 1999. [7] J.G. Sommer, D.A. Meyer, Factors controlling the dynamic properties of elastomeric products, SAE Paper No. 730267, 1973. [8] M.J. Wang, Effect of polymer-filler and filler-filler interactions on dynamic properties of filled vulcanizates, Rubber Chem. Technol. 71 (3) (1998) 520–589. [9] G. Kraus, Mechanical losses in carbon black filled rubbers, J. Appl. Polym. Sci. 39 (1984) 75–92. [10] J.D. Ulmer, Strain dependence of dynamic mechanical properties of carbon black filled rubber compounds, Rubber Chem. Technol. 69 (1995) 15–47. [11] A. Lion, Phenomenological modelling of strain-induced structural changes in filler-reinforced elastomers—a time domain formulation of the Kraus model, Kautsch. Gummi Kunsts. 58 (2005) 157–162. [12] M.J. Gregory, Dynamic properties of rubber in automotive engineering, Elastomerics 117 (11) (1985) 19–24. [13] V.A. Coveney, D.E. Johnsson, D.M. Turner, A triboelastic model for the cyclic behavior of filled vulcanizates, Rubber Chem. Technol. 68 (1995) 660–670. [14] M. Kaliske, H. Rothert, Constitutive approach to rateindependent properties of filled elastomers, Int. J. Solids Struct. 35 (17) (1998) 2057–2071.
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[15] S. Bruni, A. Collina, Modelling of viscoelastic behaviour of elastomeric components: an application to the simulation of train–track interaction, Vehicle Syst. Dyn. 34 (2000) 283–301. [16] A.K. Olsson, P.E. Austrell, A fitting procedure for a viscoelastic–elastoplastic material model, in: Second European Conference on Constitutive Models for Rubber, 2001. [17] C.R. Brackbill, G.A. Lesieutre, E.C. Smith, L.E. Ruhl, Characterization and modelling of the low strain amplitude and frequency dependent behavior of elastomeric damper materials, J. Am. Helicopter Soc. 45 (1) (2000) 34–42. [18] P.E. Austrell, A.K. Olsson, M. Jonsson, A method to analyse the non-linear dynamic behaviour of rubber components using standard FE codes, Paper no. 464, Conference on Fluid and Solid Mechanics, 2001. [19] A.K. Olsson, P.E. Austrell, Finite element analysis of a rubber bushing considering rate and amplitude dependence effects, in: Third European Conference on Constitutive Models for Rubber, 2003. [20] A.K. Olsson, P.E. Austrell, G. Sandberg, Approximate viscoelastic FE procedures in frequency and time domain to account for the Fletcher–Gent effect, in: Fourth European Conference on Constitutive Models for Rubber, 2005. [21] N. Gil-Negrete, On the modelling and dynamic stiffness prediction of rubber isolators, University of Navarra, Spain, 2004.
[22] M. Berg, A non linear rubber spring model for rail vehicle dynamics analysis, Vehicle Syst. Dyn. 30 (1998) 197–212. [23] M. Sjo¨berg, L. Kari, Non-linear behavior of a rubber isolator system using fractional derivatives, Vehicle Syst. Dyn. 37 (2002) 217–236. [24] K. Misaji, S. Hirose, K. Shibata, Vibration analysis of rubber vibration isolators of vehicle using the restoring force model of power function type, JSME Int. J. C Dyn. Con. 38 (4) (1995) 679–685. [25] M.J. Garcı´ a Ta´rrago, L. Kari, J. Vin˜olas, N. Gil-Negrete, Torsion stiffness of a rubber bushing: a simple engineering design formula including amplitude dependence, J. Strain Anal. Eng. Des. 42 (1) (2007) 13–21. [26] M.J. Garcı´ a Ta´rrago, L. Kari, J. Vin˜olas, Axial stiffness of carbon black filled rubber bushings including frequency and amplitude dependence, Kautsch. Gummi Kunsts. 60 (01/02) (2007) 43–48. [27] J.M. Horton, M.J.C. Gover, G.E. Tupholme, Stiffness of rubber bush mountings subjected to radial loading, Rubber Chem. Technol. 73 (2000) 253–264. [28] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974. [29] L. RA˚de, B. Westergren, Mathematics Handbook for Science and Engineering, Studentlitteratur, Lund, 1988.
POLYMER TESTING Polymer Testing 26 (2007) 629–638 www.elsevier.com/locate/polytest
Data Prediction
Frequency and amplitude dependence of the axial and radial stiffness of carbon-black filled rubber bushings M.J. Garcı´ a Ta´rragoa,b,, L. Karia, J. Vinolasb, N. Gil-Negreteb a
The Marcus Wallenberg Laboratory for Sound and Vibration Research, Department of Aeronautical and Vehicle Engineering, KTH, 100 44 Stockholm, Sweden b Department of Applied Mechanics, Ceit and Tecnun (University of Navarra), Manuel de Lardiza´bal 15, 20018 San Sebastia´n, Spain Received 6 February 2007; accepted 26 March 2007
Abstract The frequency and amplitude dependent dynamic behavior of carbon-black filled rubber bushings is experimentally investigated for a commercially available bushing in the axial and radial directions. Based on measurement observations, models for the axial and radial dynamic stiffness of rubber bushings are developed. The amplitude dependence—referred to as the Fletcher–Gent effect and mainly caused by the presence of carbon-black fillers in the rubber—is included in the analytical models by means of equivalent shear moduli, which result from applying a separable elastic, viscoelastic and friction material model to equivalent strains of the non-homogeneous strain states inside the bushing when subjected to axial or radial deflections. Good correlations between measurements and the axial and radial models at amplitudes of 0.1, 0.2 and 0.5 mm from 5 to 155 Hz—when the material parameters are achieved from axial measurements at 0.1 mm—prove the accuracy of both stiffness models. r 2007 Elsevier Ltd. All rights reserved. Keywords: Rubber bushing; Carbon black; Fletcher–Gent effect; Equivalent shear modulus; Dynamic stiffness
1. Introduction Rubber is a versatile polymer employed in many vehicle applications, for instance its elasticity and inherent damping make rubber bushings interesting for use in automotive suspensions. Furthermore, for engineering applications the rubber usually contains fillers, generally one of the many kinds of carbon black. This results in amplitude dependence of the dynamic behavior of the rubber—referred to as the Fletcher–Gent effect [1]. An intact filler structure Corresponding author. Tel.: +34 943 212800; fax: +34 943 213076. E-mail address: [email protected] (M.J. Garcı´ a Ta´rrago).
0142-9418/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2007.03.011
displays a large shear modulus magnitude for small amplitudes due to the interactions within the filler and between the filler and the rubber matrix, but as the amplitude increases the structure links break resulting in modulus decrease. Furthermore, the breaking of filler structure, described as frictional behavior, initially increases the loss factor, but as the amplitude increases further the frictional behavior is released, resulting in a low loss factor. This nonlinear behavior, extensively investigated by Medalia [2,3], Payne [4], Dean et al. [5], Jurado et al. [6], Sommer and Meyer [7] and Wang [8], has been characterized by several authors. Some of them developed micromechanically motivated material models, such as Kraus [9] who explains the
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amplitude dependence as due to the continuous breaking and reforming of van der Waals bonds between carbon-black aggregates. Modifications and experimental application were carried out by Ulmer [10] and Lion [11] presenting a time domain formulation of the Kraus model. Other authors use frictional models, such as Gregory [12], Coveney et al. [13] and Kaliske and Rothert [14] proposing the Prandtl element, a Coulumb damper in series with an elastic spring; a model expanded by Bruni and Collina [15], Olsson and Austrell [16] and Brackbill et al. [17]. Dynamic measurements performed on a commercially available rubber bushing reveal that the component presents frequency and amplitude dependence similar to that of the shear modulus of carbon-black filled rubber [3]. However, the previously described material models cannot be directly applied when modelling the dynamic stiffness of rubber bushings because the strain state inside the deformed components is not homogeneous. Some authors employ material models with finite element simulations and calculate the dynamic stiffness after long-time overlay procedures, such as Austrell et al. [18–20] who represent frequency and amplitude dependence by adding integer derivatives to stickslip friction components, similar to Gil-Negrete [21] except for the use of fractional derivatives. On the other hand, other authors work directly at the component level, like Berg [22] who presents a fiveparameter model that gives good resemblance to the rubber frictional behavior, also used by Sjo¨berg and Kari [23] together with a rate-dependent part using fractional derivatives. Misaji et al. [24] took into account amplitude dependence with the parameters of an ordinary Kelvin–Voigt model updated continuously for every oscillation cycle. However, the latter methods neglect the complicated amplitude dependence inside the rubber bushing due to nonhomogeneous strain states. This paper verifies the accuracy of novel engineering models representing the dynamic behavior of carbon-black filled rubber bushings in axial and radial directions while including frequency and amplitude dependence. A novel model for the radial stiffness is developed here while the formula for the axial stiffness—also applied in this paper—was previously introduced in Garcı´ a Ta´rrago et al. [26]. The analytical models are based on measurement observations and depend on the bushing geometry and an equivalent shear modulus. The latter is obtained by applying a separable elastic,
viscoelastic and friction material model [25,26] to an equivalent strain of the non-homogeneous strain state inside the bushing when subjected to axial or radial deflections. The good agreement between measured and modelled stiffness in axial and radial directions at amplitudes of 0.1, 0.2 and 0.5 mm over a frequency range from 5 to 155 Hz when the model parameters are achieved from axial measurements at 0.1 mm proves the accuracy of the stiffness models. 2. Experimental The frequency and amplitude dependence of carbon-black filled rubber bushings was experimentally investigated for a commercially available bushing. Measurements were carried out in a servo-hydraulic testing machine. The test object in Fig. 1, with mean length L ¼ 34 mm, and inner and outer radii a ¼ 10 mm and b ¼ 18 mm, respectively, was manufactured by Caucho Metal Productos (CMP) under the trade name 15 961 with rubber hardness, Shore A of 61 2. The vulcanized compound contains NR (natural rubber) and 27 phr (parts per hundred rubber) of reinforcing carbon-black filler. 2.1. Setup Experiments were conducted in three stages, see Fig. 2. Firstly, the test object was mounted on a Schenck Series 56 servo-hydraulic test machine whose main components are the frame, an actuator and a hydraulic power pack. The frame consists of the smooth standard platen, columns and the crosshead. The actuator, mounted on the crosshead, is a double-acting force generator, having equal piston areas in the two cylinder chambers. Inside
Fig. 1. Rubber bushing manufactured by Caucho Metal Productos (CMP) under the trade name 15961.
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SCHENCK SERVO-HYDRAULIC TEST MACHINE
LOAD F FEEDBACK SIGNAL PROCESSOR
SENSOR
ADC
DISPL.
d
SENSOR
ADC
RUBBER BUSHING
i
DIALOGUE PROCESSOR
CONTROL PROCESSOR
DAC
PISTON
SERIAL INTERFACE
INTERFACE
F : Force d : Displacement i : Current ADC : Analog-to-digital converter DAC : Digital-to-analogconverter
Fig. 2. Measurement setup.
each piston, one inductive displacement transducer measures the piston position and the piston displacement. The actuator force corresponds to the difference in the oil pressure of the two actuator chambers, measured by pressure transducers based on a strain gauge measuring system. The resulting force in the bushing is measured by a load sensor. Next, measured force and displacement data are sent to a Hydropuls electronic unit Series 56 control panel—which can constantly digitize up to five measuring channels. The signal to the actuator is then automatically adjusted in a control loop with regard to the control displacement signal. Finally,
the data are processed and transmitted to the PC where the frequency signals are displayed. No external analyzer is needed. 2.2. Measurements Dynamic measurements were performed on the test object at displacement amplitudes of 0.1, 0.2 and 0.5 mm in axial and radial directions. The bushing in Fig. 1 was excited by stepped sine displacements starting at 5 Hz and increasing with a constant frequency step of 10 Hz to a maximum frequency of 155 Hz, with amplitude held constant.
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632
Magnitude (N/m)
8
Axial
x 105
x 105
8
6
6
4
4 0.1mm 0.2mm 0.5mm
2
25 Hz 75 Hz 115 Hz
2
0
0 50
100
150
1
2
3
4
5
Loss Factor
x 10-4 0.15
0.15
0.1
0.1
0.05
0.1 mm 0.2 mm 0.5 mm
25 Hz 75 Hz 115 Hz
0.05
0
0 50 100 Frequency (Hz)
150
1
2
3 4 Amplitude (m)
5 x
10-4
Fig. 3. Left: Frequency dependence of measured axial stiffness at three amplitudes. Right: Amplitude dependence of measured axial stiffness at three frequencies.
The resulting complex stiffness in Figs. 3 and 4 is expressed by a magnitude and a loss factor—defined as the tangent of the loss angle. Results reveal that the stiffness presents frequency and amplitude dependence similar to that of the shear modulus of carbon-black filled rubber. The magnitude and loss factor increase with frequency, as shown on the left of Figs. 3 and 4, while the magnitude takes high values for small amplitudes, but as the amplitude increases the magnitude decreases. Furthermore, the loss factor initially increases but as the amplitude increases further it decreases, as displayed on the right of Figs. 3 and 4. The drop of the radial loss factor—measured at 0.5 mm of amplitude—over the frequency range is mainly attributed to the difficulties in accurately measuring amplitudes higher than 0.2 mm. Finally, quasi-static measurements were performed on the servo-hydraulic Schenck testing machine using the same setup. 3. Model Based on measurement observations, analytical models for the axial and radial stiffness were
developed. The former was already presented in Garcı´ a Ta´rrago et al. [26] while a novel radial model is expounded here. They are functions of the bushing dimensions and an equivalent shear modulus. The latter was obtained by applying a newly developed rubber material model to an equivalent strain of the non-homogeneous strain state inside the rubber bushing when subjected to axial or radial deformations. 3.1. Rubber model The rubber material model recently developed in Garcı´ a Ta´rrago et al. [25] is the result of applying the force–displacement relation of the rubber component model established by Sjo¨berg and Kari [23] to the stress–strain level. It consists of three parallel components which represent the elasticity, viscoelasticity and amplitude dependence by using only five parameters. Firstly, in Fig. 5 the elasticity is modelled by a linear relation between the elastic stress selast and strain ; selast ðtÞ ¼ 2mðtÞ,
(1)
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Radial
Magnitude (N/m)
x 106 6
6
4
4
2
x 106
2
0.1mm 0.2mm 0.5mm
0
633
25 Hz 75 Hz 115 Hz
0 50
100
1
150
2
3
4
5
Loss Factor
x 10-4
0.1
0.1
0.1 mm 0.2 mm 0.5 mm
0.05
0
25 Hz 75 Hz 115 Hz
0.05
0 50 100 Frequency (Hz)
150
1
2
3 4 Amplitude (m)
5 x 10-4
Fig. 4. Left: Frequency dependence of measured radial stiffness at three amplitudes. Right: Amplitude dependence of measured radial stiffness at three frequencies.
σtotal (t)
where sfract is the viscoelastic stress and Da denotes the fractional time derivative of order a, defined through an analytical continuation of a fractional Riemann–Liouville integration [28]. Numerically, the viscoelastic component is evaluated as
ε (t)
m, α 2μ
σf max, ε1/2
sfract ðtÞ m
Fig. 5. Rubber material model.
where m is the elastic shear modulus and t is time. The second branch in Fig. 5 indicates the frequency dependence and is modelled by using fractional derivatives which increase the ability to fit to measured frequency dependent characteristics while keeping the number of parameters to only two: a proportionality constant m and the time derivative order a; sfract ðtÞ ¼ mDa ðtÞ;
0oap1,
(2)
n1 Dta X Gðj aÞ nj , GðaÞ j¼0 Gðj þ 1Þ
(3)
where nj ¼ ððn jÞ DtÞ, n is the final strain at time tn ¼ n Dt, Dt is a constant time step applied in the estimation process and G denotes the Gamma function [29] defined as Z 1 1 X ð1Þj , GðbÞ ¼ xb1 ex dx þ j!ðb þ jÞ 1 j¼0 ba0; 1; 2; . . . .
ð4Þ
The third branch in the rubber model represents the amplitude dependence by a smooth friction component [22] which enables a very good fit to measured curves using only two parameters sfmax and 1=2 . The frictional stress sfrict develops gradually
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following the equation: sfrict ðtÞ ¼ sfs þ
½ðtÞ s ½sfmax signð_Þsfs , 1=2 ½1 signð_Þsfs =sfmax þ signð_Þ½ðtÞ s
(5) where the material parameters sfmax and 1=2 are the maximum possible friction stress and the strain needed to develop half of that stress, respectively, with signð_Þ denoting the sign of the strain rate. The values of sfs and s are updated each time the strain changes direction at _ ¼ 0 as sfs sfrict j_¼0 and s j_¼0 . Finally, the rubber behavior is represented by a total stress which is the sum of the three stresses: stotal ðtÞ ¼ selast ðtÞ þ sfract ðtÞ þ sfrict ðtÞ.
(6)
However, a shear modulus cannot be directly achieved from the non-linear relation in Eq. (6) between the total stress and strain, as would be the case with the elastic component, Eq. (1). Moreover, that relation is valid at that specific strain, but in order to represent the behavior of the whole bushing a ‘global’ value for the shear modulus is required. A simplified and effective solution consists in calculating an equivalent shear strain of the nonhomogeneous strain state inside the bushing by using the classical theory of elasticity. Next, the rubber material model is applied to it, thus yielding an equivalent shear modulus which is inserted into an analytical formula for the bushing stiffness.
and d a the amplitude. Similarly, a harmonic radial displacement d r ðtÞ ¼ d r sinðo0 tÞ applied at the outer surface of the bushing results in an equivalent strain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 equiv r ðtÞ ¼ d r ðtÞ . 4 ½b a4 loge ðb=aÞ ½a2 b2 2 (8) 3.3. Stiffness formulas The separable elastic, viscoelastic and friction material model presented in Section 3.1 is applied to the time-domain equivalent strains in Eqs. (7) and (8), thus yielding total stresses stotal a=r ðtÞ of the form displayed in Eq. (6). Next, an equivalent shear modulus is obtained by dividing the temporal Fourier transform (~) of the total stress s~ total a=r ðoÞ by the corresponding equivalent strain ~ equiv a=r ðoÞ Axial 500 400 300 200 Force (N)
634
100 0 -100 -200
3.2. Equivalent strains
-300
The strain states inside the rubber bushing in Fig. 1 when subjected to axial and radial deflections are obtained with the classical linear theory of elasticity while considering that the constitutive equations between the stress and strain contain only elastic and frequency dependent components, and that the wave effects within the bushing are negligible for the frequency range of interest. The amplitude dependence is taken into account later when the rubber model is applied to the equivalent strains. An energy density balance [25] between the rubber bushing deformed axially and a simple shear specimen made of the same material leads to the axial equivalent strain d a ðtÞ equiv a ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 2½b a2 loge ðb=aÞ
(7)
where d a ðtÞ ¼ d a sinðo0 tÞ is the harmonic axial displacement with o0 the angular excitation frequency
Model
-400
Measurement
-500 -1
-0.5
0 Displacement (m)
0.5
1 x
10-3
Fig. 6. Quasi-static measured and modelled hysteresis loop at amplitude of 1 mm and frequency of 0.1 Hz.
Table 1 Optimized material parameters for the bushing in Fig. 1 obtained from axial measurements Material parameters m ¼ 1:09 106 N=m2 m ¼ 2:0 105 Nsa =m2 a ¼ 0:24 sf max ¼ 1:46 104 N=m2 1=2 ¼ 1:3 102
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x 105
evaluated at the excitation frequency o0 ,
Axial
Magnitude (N/m)
8
m^ equiv
6
Mod. 0.1mm Meas. 0.1mm
2 0 40
60
80
100
120
140
Loss Factor
0.15 0.1
k^dyn
0.05
Mod. 0.1mm Meas. 0.1mm 40
60 80 100 Frequency (Hz)
120
a
¼
140
s~ total a=r ðo0 Þ . 2 ~equiv a=r ðo0 Þ
(9)
2pL m^ , loge ðb=aÞ equiv a
(10)
10pL m^ equiv r , 2 2 2 2 2 loge ðb=aÞ ½ðb a =b þ a Þ þ D
k^dyn r ¼ 7
Fig. 7. Measured and modelled axial stiffness versus frequency at amplitude of 0.1 mm.
6
3 2
(11) Axial
x 105
Magnitude (N/m)
¼
while the radial equivalent modulus is inserted into the formula for the radial stiffness of a bushing of finite length developed by Horton et al. [27]:
0 20
a=r ðo0 Þ
This linearization process takes into account the non-linear relation between stress and strain at frequency o0 considering the first order response while omitting the less important overtones (stress response at 3o0 , 5o0 ; . . . ). Finally, the axial equivalent modulus is inserted into the formula for the axial stiffness calculated previously with the classical theory of elasticity
4
20
635
x 105
6
4
4 Mod. 0.2mm Meas. 0.2mm Mod. 0.5mm Meas. 0.5mm
2
Mod. 45 Hz Meas. 45 Hz Mod. 115 Hz Meas. 115 Hz
2
0
0 50
100
150
1
2
3
4
5
Loss Factor
x 10-4
0.15
0.15
0.1
0.1
0.05
Mod. 0.2mm Meas. 0.2mm Mod. 0.5mm Meas. 0.5mm
Mod. 45 Hz Meas. 45 Hz Mod. 115 Hz Meas. 115 Hz
0.05
0
0 50 100 Frequency (Hz)
150
1
2
3 Amplitude (m)
4
5 x 10-4
Fig. 8. Left: Measured and modelled axial stiffness versus frequency at two amplitudes. Right: Measured and modelled axial stiffness versus amplitude at two frequencies.
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section to verify the accuracy of Eqs. (10) and (11) modelling the axial and radial stiffness of carbonblack filled rubber bushings. Firstly, the model parameters corresponding to the analyzed bushing are determined. The dimensions of the bushing are straightforwardly obtained while the material parameters—common to both models—are extracted only from axial tests. The elastic and amplitude dependence parameters m, sfmax and 1=2 are those which optimize the agreement between measured and modelled hysteresis loops for the axial case at amplitude of 1 mm and frequency of 0.1 Hz, as shown in Fig. 6. The frequency dependence parameters m and a are obtained from a minimization of the errors between the axial model and measurements at amplitude of 0.1 mm over a frequency range from 5 to 155 Hz. The optimized material parameters values are in Table 1 with the corresponding stiffness displayed in Fig. 7. Next, the axial model—using the parameters values in Table 1—is carried out at amplitudes of 0.2 and 0.5 mm from 5 to 155 Hz and the results are
with 4ðb2 þ a2 Þ abðb2 þ 3a2 Þ½I 1 ðabÞK 0 ðaaÞ þ I 0 ðaaÞK 1 ðabÞ D¼ a2 baðb2 þ a2 Þ½I 1 ðabÞK 1 ðaaÞ þ I 1 ðaaÞK 1 ðabÞ
aað3b2 þ a2 Þ½I 1 ðaaÞK 0 ðabÞ þ I 0 ðabÞK 1 ðaaÞ , a2 baðb2 þ a2 Þ½I 1 ðabÞK 1 ðaaÞ þ I 1 ðaaÞK 1 ðabÞ
ð12Þ where I n ðarÞ and K n ðarÞ are modified Bessel functions of the first and second kinds, respectively, and of order n [29], r is the radius and a2 ¼ 60=L2 . These engineering formulas represent the dynamic behavior including frequency and amplitude dependence of a carbon-black filled rubber bushing in axial and radial directions. The complex stiffness is expressed by a magnitude and a loss factor, the latter defined as the tangent of the phase between stress and strain at the excitation frequency. 4. Results and discussions Measurements performed on the commercially available bushing in Fig. 1 are employed in this
Radial
Magnitude (N/m)
x 106
x 106
5
5
4
4
3
3 Mod. 0.1mm Meas. 0.1mm Mod. 0.2mm Meas. 0.2mm
2 1
Mod. 25 Hz Meas. 25 Hz Mod. 75 Hz Meas. 75 Hz
2 1 0
0 50
100
150
1
2
3
4
5 x 10-4
Loss Factor
0.15
0.1
0.1
0.05
Mod. 0.1mm Meas. 0.1mm Mod. 0.2mm Meas. 0.2mm
Mod. 25 Hz Meas. 25 Hz Mod. 75 Hz Meas. 75 Hz
0.05
0
0 50 100 Frequency (Hz)
150
1
2
3 4 Amplitude (m)
5 x 10-4
Fig. 9. Left: Measured and modelled radial stiffness versus frequency at two amplitudes. Right: Measured and modelled radial stiffness versus amplitude at two frequencies.
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compared with the dynamic measurements presented in Section 2, showing good agreement, as seen in Fig. 8. Finally, the radial model—making use of identical parameter values as previously employed in the axial model—is performed at amplitudes of 0.1, 0.2 and 0.5 mm and the results are displayed together with the experimental data, showing the good agreement presented in Fig. 9. Consequently, the effectiveness of Eqs. (10) and (11) modelling the global behavior of filled rubber bushings including frequency and amplitude dependence is verified.
637
Unlike previous models, these formulas implemented in MATLABs are simple and fast engineering tools to predict the dynamic stiffness of rubber bushings including frequency and amplitude dependence. The simplicity—characteristic of these models—is important when a rubber bushing is only a small component in more complex systems where the dynamic behavior is to be analyzed. Acknowledgment Caucho Metal Productos (CMP) is gratefully acknowledged for providing the sample bushings.
5. Conclusions This paper experimentally verifies the accuracy of novel engineering models representing the dynamic behavior of carbon-black filled rubber bushings in axial and radial directions, while including frequency and amplitude dependence. The model for the radial stiffness is developed in this paper while the axial stiffness model was previously introduced in Garcı´ a Ta´rrago et al. [26]. Experiments were carried out on a commercially available bushing. Good agreement between measured and modelled axial and radial stiffness at amplitudes of 0.1, 0.2 and 0.5 mm over a frequency range from 5 to 155 Hz—when the material parameters are achieved from axial measurements at 0.1 mm—prove the accuracy of both stiffness models. Therefore, these engineering formulas together with the torsion model developed in Garcı´ a Ta´rrago et al. [25] represent accurately enough the global dynamic behavior of filled rubber bushings including frequency and amplitude dependence. The developed analytical models are based on observations of the measurements performed on the carbon-black filled rubber bushing, showing that the frequency and amplitude dependence of its dynamic stiffness is similar to that dependence of the shear modulus of that type of rubber. Thus, the engineering formulas are function of the bushing dimensions and an equivalent shear modulus. The latter is achieved by applying a separable elastic, viscoelastic and friction material model to an equivalent strain of the non-homogeneous strain state inside the bushing when deformed. The rubber model is the result of applying a force–displacement relation of a well established rubber component model [23]—experimentally validated on commercial bushings—to the stress–strain level.
References [1] W.P. Fletcher, A.N. Gent, Non-linearity in the dynamic properties of vulcanised rubber compounds, I.R.I. Transactions 29 (1953) 266–280. [2] A.I. Medalia, G. Kraus, Reinforcement of elastomers by particulate fillers, in: J.E. Mark, B. Erman, F.R. Eirich (Eds.), Science and Technology of Rubber, Academic Press, San Diego, 1994, pp. 387–418. [3] A.I. Medalia, Effects of carbon black on dynamic properties of rubber, Rubber Chem. Technol. 51 (1978) 437–523. [4] A.R. Payne, R.E. Whittaker, Low strain dynamic properties of filled rubbers, Rubber Chem. Technol. 44 (1971) 440–478. [5] G.D. Dean, J.C. Duncan, A.F. Johnson, Determination of nonlinear dynamic properties of carbon-filled rubbers, Polym. Test. 4 (2–4) (1984) 225–249. [6] F.J. Jurado, A. Mateo, N. Gil-Negrete, J. Vinolas, L. Kari, Testing and FE modelling of the dynamic properties of carbon black filled rubber, in: Proceeding of EAEC 99, Barcelona, 1999. [7] J.G. Sommer, D.A. Meyer, Factors controlling the dynamic properties of elastomeric products, SAE Paper No. 730267, 1973. [8] M.J. Wang, Effect of polymer-filler and filler-filler interactions on dynamic properties of filled vulcanizates, Rubber Chem. Technol. 71 (3) (1998) 520–589. [9] G. Kraus, Mechanical losses in carbon black filled rubbers, J. Appl. Polym. Sci. 39 (1984) 75–92. [10] J.D. Ulmer, Strain dependence of dynamic mechanical properties of carbon black filled rubber compounds, Rubber Chem. Technol. 69 (1995) 15–47. [11] A. Lion, Phenomenological modelling of strain-induced structural changes in filler-reinforced elastomers—a time domain formulation of the Kraus model, Kautsch. Gummi Kunsts. 58 (2005) 157–162. [12] M.J. Gregory, Dynamic properties of rubber in automotive engineering, Elastomerics 117 (11) (1985) 19–24. [13] V.A. Coveney, D.E. Johnsson, D.M. Turner, A triboelastic model for the cyclic behavior of filled vulcanizates, Rubber Chem. Technol. 68 (1995) 660–670. [14] M. Kaliske, H. Rothert, Constitutive approach to rateindependent properties of filled elastomers, Int. J. Solids Struct. 35 (17) (1998) 2057–2071.
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[15] S. Bruni, A. Collina, Modelling of viscoelastic behaviour of elastomeric components: an application to the simulation of train–track interaction, Vehicle Syst. Dyn. 34 (2000) 283–301. [16] A.K. Olsson, P.E. Austrell, A fitting procedure for a viscoelastic–elastoplastic material model, in: Second European Conference on Constitutive Models for Rubber, 2001. [17] C.R. Brackbill, G.A. Lesieutre, E.C. Smith, L.E. Ruhl, Characterization and modelling of the low strain amplitude and frequency dependent behavior of elastomeric damper materials, J. Am. Helicopter Soc. 45 (1) (2000) 34–42. [18] P.E. Austrell, A.K. Olsson, M. Jonsson, A method to analyse the non-linear dynamic behaviour of rubber components using standard FE codes, Paper no. 464, Conference on Fluid and Solid Mechanics, 2001. [19] A.K. Olsson, P.E. Austrell, Finite element analysis of a rubber bushing considering rate and amplitude dependence effects, in: Third European Conference on Constitutive Models for Rubber, 2003. [20] A.K. Olsson, P.E. Austrell, G. Sandberg, Approximate viscoelastic FE procedures in frequency and time domain to account for the Fletcher–Gent effect, in: Fourth European Conference on Constitutive Models for Rubber, 2005. [21] N. Gil-Negrete, On the modelling and dynamic stiffness prediction of rubber isolators, University of Navarra, Spain, 2004.
[22] M. Berg, A non linear rubber spring model for rail vehicle dynamics analysis, Vehicle Syst. Dyn. 30 (1998) 197–212. [23] M. Sjo¨berg, L. Kari, Non-linear behavior of a rubber isolator system using fractional derivatives, Vehicle Syst. Dyn. 37 (2002) 217–236. [24] K. Misaji, S. Hirose, K. Shibata, Vibration analysis of rubber vibration isolators of vehicle using the restoring force model of power function type, JSME Int. J. C Dyn. Con. 38 (4) (1995) 679–685. [25] M.J. Garcı´ a Ta´rrago, L. Kari, J. Vin˜olas, N. Gil-Negrete, Torsion stiffness of a rubber bushing: a simple engineering design formula including amplitude dependence, J. Strain Anal. Eng. Des. 42 (1) (2007) 13–21. [26] M.J. Garcı´ a Ta´rrago, L. Kari, J. Vin˜olas, Axial stiffness of carbon black filled rubber bushings including frequency and amplitude dependence, Kautsch. Gummi Kunsts. 60 (01/02) (2007) 43–48. [27] J.M. Horton, M.J.C. Gover, G.E. Tupholme, Stiffness of rubber bush mountings subjected to radial loading, Rubber Chem. Technol. 73 (2000) 253–264. [28] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974. [29] L. RA˚de, B. Westergren, Mathematics Handbook for Science and Engineering, Studentlitteratur, Lund, 1988.